1 Introduction
Let us assume that
A and
B are two nonempty subsets of a metric space
\((X,d)\) and
\(T:A\longrightarrow B\). Clearly
\(T(A)\cap A\neq \emptyset\) is a necessary condition for the existence of a fixed point of
T. The idea of the best proximity point theory is to determine an approximate solution
x such that the error of equation
\(d(x,Tx)=0 \) is minimum. A solution
x for the equation
\(d(x,Tx)=d(A,B)\) is called a best proximity point of
T. The existence and convergence of best proximity points have been generalized by several authors [
1‐
8] in many directions. Also, Akbar and Gabeleh [
9,
10], Sadiq Basha [
11] and Pragadeeswarar and Marudai [
12] extended the best proximity points theorems in partially ordered metric spaces (see also [
13‐
18]). On the other hand, Suzuki [
19] introduced the concept of
τ-distance on a metric space and proved some fixed point theorems for various contractive mappings by
τ-distance. In this paper, by using the concept of
τ-distance, we prove some best proximity point theorems.
2 Preliminaries
Let
A,
B be two nonempty subsets of a metric space
\((X,d)\). The following notations will be used throughout this paper:
$$\begin{aligned}& d(y,A):=\inf\bigl\{ d(x,y):x\in A\bigr\} , \\& d(A,B):=\inf\bigl\{ d(x,y):x\in A\mbox{ and }y\in B \bigr\} , \\& A_{0} :=\bigl\{ x \in A : d(x, y)= d(A, B)\mbox{ for some }y \in B \bigr\} , \\& B_{0} :=\bigl\{ y \in B : d(x, y)= d(A, B)\mbox{ for some }x \in A \bigr\} . \end{aligned}$$
We recall that
\(x\in A\) is a best proximity point of the mapping
\(T:A\longrightarrow B\) if
\(d(x,Tx)=d(A,B)\). It can be observed that a best proximity point reduces to a fixed point if the underlying mapping is a self-mapping.
It is clear that, for any nonempty subset A of X, the pair \((A,A)\) has the P-property.
Samet
et al. [
21] introduced a class of contractive mappings called
α-
ψ-contractive mappings. Let Ψ be the family of nondecreasing functions
\(\psi:[0,\infty )\longrightarrow[0,\infty)\) such that
\(\sum_{n=1}^{\infty}\psi ^{n}(t)<\infty\) for all
\(t>0\), where
\(\psi^{n}(t) \) is the
nth iterate of
ψ.
In the following examples, we define \(\eta:X \times[0,\infty )\longrightarrow[0,\infty)\) by \(\eta(x,t)= t\) for all \(x\in{X}\), \(t\in [0,\infty )\). It is easy to see that p is a τ-distance on a metric space X.
The following lemmas are essential for the next sections.
The next result is an immediate consequence of Lemma
2.14 and Lemma
2.16.
3 Some best proximity point theorems
Now, we define the weak P-property with respect to a τ-distance as follows.
It is clear that, for any nonempty subset A of X, the pair \((A,A)\) has the weak P-property with respect to p.
It is easy to see that if \((A,B)\) has the P-property, then \((A,B)\) has the weak P-property.
By the definition of
A and
B, we obtain
$$d \bigl((0,2),(1,1) \bigr)=d \bigl((0,3),(1,4) \bigr)=d(A, B)=\sqrt{2}, $$
where
\((0,2),(0,3)\in A \) and
\((1,1),(1,4)\in B\). We have
$$\begin{aligned}& p_{1} \bigl((0,2),(0,3) \bigr)=5\quad \mbox{and} \quad p_{1} \bigl((1,1),(1,4) \bigr)=\sqrt {2}+\sqrt{17}, \\& p_{1} \bigl((0,3),(0,2) \bigr)=5 \quad\mbox{and} \quad p_{1} \bigl((1,4),(1,1) \bigr)=\sqrt {17}+\sqrt{2}. \end{aligned}$$
Therefore
\((A,B) \) has the weak
P-property with respect to
\(p_{1}\). On the other hand, we have
$$p_{2} \bigl((0,3),(0,2) \bigr)=2 \quad\mbox{and}\quad p_{2} \bigl((1,4),(1,1) \bigr)=\sqrt{2}. $$
This implies that
\((A,B) \) has not the weak
P-property with respect to
\(p_{2}\).
We first prove the following lemma. Then we state our results.
The following result is a special case of Lemma
3.8 obtained by setting
α defined in Remark
2.6.
The following result is a spacial case of Lemma
3.8 if
g is the identity map.
The following result is the special case of Theorem
3.11 obtained by setting
\(p=d\).
The next result is an immediate consequence of Theorem
3.13 by taking
\(A=B\) and
\(p=d\).
4 α-p-Proximal contractions
It is easy to see that
$$d \bigl(-2,T(-3) \bigr)=d \bigl(2,T(3) \bigr)=d(A,B)=1. $$
If
\(r\in[\frac{2}{3},1) \), then we have
$$\begin{aligned}& p(-2,2)\leq rp(-3,3), \\& p(2,-2)\leq rp(3,-3). \end{aligned}$$
Hence
T is a
p-proximal contraction of the first kind. Also,
$$\begin{aligned}& p \bigl(T(-2),T(2) \bigr)\leq rp \bigl(T(-3),T(3) \bigr), \\& p \bigl(T(2),T(-2) \bigr)\leq rp \bigl(T(3),T(-3) \bigr) \end{aligned}$$
for all
\(r\in[0,1) \). This implies that
T is a
p-proximal contraction of the second kind.
It is easy to see that
$$d \bigl(-2,T(-2) \bigr)=d \bigl(2,T(3) \bigr)=d(A,B)=1 \quad\mbox{and}\quad {-}2 \preceq3 . $$
If
\(r\in[\frac{2}{3},1) \), then we have
$$p(-2,2)\leq rp(-2,3) . $$
\(p(2,-2)\nleq rp(3,-2)\), but it is not necessary because
\(3\npreceq -2 \). Hence
T is an ordered
p-proximal contraction of the first kind. But
T is not a
p-proximal contraction of the first kind because
\(p(2,-2)\nleq rp(3,-2)\) for all
\(r\in[0,1)\). Also,
$$p \bigl(T(-2),T(2) \bigr)\leq rp \bigl(T(-2),T(3) \bigr) $$
for all
\(r\in[0,1) \). Notice that
\(p (T(2),T(-2) )\nleq rp (T(3),T(-2) ) \), but it is not necessary because
\(3\npreceq-2 \). This implies that
T is an ordered
p-proximal contraction of the second kind. But
T is not a
p-proximal contraction of the second kind because
\(p (T(2),T(-2) )\nleq rp (T(3),T(-2) )\) for all
\(r\in[0,1)\).
The next result is an immediate consequence of Theorem
4.9 by setting
α defined in Remark
2.6.
The next result is obtained by taking
\(p=d\) in Theorem
4.11.
The following result is a best proximity point theorem for nonself α-p-proximal contraction of the second kind.
The next result is an immediate consequence of Theorem
4.13 by setting
α defined in Remark
2.6.
The following result is obtained by taking
\(p=d\) in Theorem
4.15.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors carried out the proof. All authors conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.